Distance between parallel lines: Examples

Solution The only thing what we need to do is transform both the given lines to the form ax + by + c = 0 (i.e. transform the equation so that RHS = 0).

Next, if both the lines have corresponding coefficients of x and y equal, we can apply the formula. Else, we’ll transform one of the equations by multiplying with some constant so that the coefficients become equal.

(i) The two lines can be written as 4x + 3y – 5 = 0 and 4x + 3y – 10 = 0. Therefore, the required distance is equal to |(-5)-(-10)|/\(\sqrt{4^2+3^2}\) = 1

(ii) The two equations can be written as x – y + 1 = 0 and 2x – 2y – 5 = 0. But before we can use the formula, we must multiply the first equation by 2, so that it becomes 2x – 2y + 2 = 0.

Now, the distance can be calculated as |2-(-5)|/\(\sqrt{2^2+(-2)^2}\) = 7/2\(\sqrt{2}\)

Example 2 The equations of two sides of a square are 3x + 4y – 5 = 0 and 3x + 4y – 15 = 0. If the third side passes through (6, 5), find the equations of the remaining sides.

Solution Note that the two given lines are parallel, and therefore are equations of opposite sides.

The other two sides will have equations of the form 4x – 3y + k = 0. (since they are perpendicular to the given sides).

Since one of them passes through (6,5) we get one value of k as -9, and therefore, one of the remaining sides as 4x – 3y – 9 = 0.

What about the other one?

Since the equations represent sides of a square, the distance between the opposite sides will be equal.